So let's start with some algebra. So
there are nice simple equations like x
plus 1 equals 5; course it's not too hard
to solve that equation, we get x equals 4.
Now this is an equation in whole numbers
and we solved it in whole numbers; pretty
soon if you write down equations you get
things you can't solve in whole numbers.
If the equation had been x plus 6 equals
5 there's no whole number solution to x,
I would need minus 1 for that so that's
x equals minus 1. So I need to introduce
negative numbers to do that. That's good,
but after all we might face an equation
like 2x plus 1 equals 0, so we need some
fractions as well to keep going on solving
these equations. After a while you begin
wondering why you didn't multiply xs
together and you get polynomial equations. So
a polynomial is something like a
constant times x to the n plus another
number times x to the n minus 1 plus - blah blah blah.
(Brady: These are equations that have got things)
(to the power?) - Right, they're not even
equations they're really functions. If I
have a function f of x then I talk- I
could solve an equation but an equation
needs an equal sign. So f of x equals 0
is an equation and I talk about a
solution, but if I have just a function f
of x then I can talk about a root of the
function, that's the same thing as a
solution. So I have have the habit of
alternating between talking about roots
and and solutions so I don't want to
confuse anybody. Let's look at an
equation, if we had x squared minus 1
equals 0, x could be plus 1, that's a
solution, but there's another solution 2x
equals minus 1. If I have other
polynomial equations though it's not so
easy to find solutions; for instance a
simple one would be x squared equals 2.
Well, there's no rational number that
solves that equation, that's something
the Greeks discovered they considered it a deep
philosophical mystery. But now of course
we just write x equals square root of 2.
And there's another one, if you'll believe
that there's a square root of 2 then you
surely believe there's a minus square
root of 2, so that's another solution. So
again I have two solutions which is
pretty nice. So I could go on that way,
unfortunately I quickly hit equations
that don't have solutions in real
numbers, something like x squared plus 1
equals 0 has no real number solutions
because if I put in a real number x this
is always has a positive square and when
I add 1 it's even more positive so I can
never get 0. So at this point is begin- nice
to begin graphing equations. For instance
if f of x is x squared plus 1 then when x
is 0 I get the value of 1, so I put a 1
there, and when x is for example 1 then
this would be 2 so I'd be up here. When x
is 2 it's 4 so it's up here somewhere;
and everybody probably has seen this as
a parabola. So it doesn't have any roots,
it doesn't have any any values of x
which make the value of f of x be 0. You
can see that from the picture because a
solution would be a place where the
graph crossed the x-axis, in other words
where y was 0, this line is where y
equals 0. So for a function to have a
root or an equation to have a solution the
graph of the function has to cross the
x-axis. - (It's like an electric fence or a)
(mag- special magic line.) - A magic line,
right. There is one kind of polynomial
equation that always has a root in real
numbers and that's the case when F has
odd degree. So every polynomial has a
degree, in this case it's n - exponent on
the highest power. If you have a
polynomial of odd degree;
well this one was of even degree and
that turns out in some way to be why it
didn't have a root. If- let's take one of
odd degree. If we took for instance the
function f of x equals x- well f of x
equals x of course is of odd degree and
the graph of that is just a straight
line that goes through the origin. You
can see it has a solution when x equals
0 but I could do fancier
things, I could say f of x equals x- maybe
x squared times x minus 1, what does that
look like? Well I can see from the way I
wrote it that f of x equals 0 has two
roots; at x equals 1 and the other one is
at x equals 0. And the graph is going to
look, it turns out, something like this.
And I'll explain how I knew that in a
second, but I see the root and I see that
there's a really robust root there. This
one, I sort of almost missed if I'd
move down just a little bit it wouldn't
have been a root anymore. For instance if
I put here minus 1 then this whole
picture would move down by 1, but it
would still have a root here. And this is
really a robust picture somehow. So that
that's there's a theorem behind that;
mathematicians love to write the word
theorem, especially when they can prove
what they're about to write. Any
polynomial, and here I mean with real
coefficients, of odd degree. - (That's not)
(the coefficients that are of odd degree, this is)
(the polynomial?) - The polynomial has odd
degree that's right; begins something like
x to the fifth plus 7x to the
fourth plus something else. -Of odd degree
has at least one real root. This odd
exponent is going to be the key, that's
right. Just from that I'm going to show
you that there's some w which when you
put it in plug it in here it gives zero
as the answer. That's pretty amazing that
you can say that just from knowing that
little piece of the polynomial. Anyway
this is a proof that really explains
something I think. I'm going to graph the
polynomial, you might ask me how can you
graph the polynomial if you don't know
anything about it? But I'm going to do it
anyway. I'm going to assume that a is
positive just for the sake of argument,
to make make one picture, but there would
be an exactly corresponding picture if a
were negative. And I want to find a value
of x where y is zero, and of course that
means a place where the graph will cross
the x axis. The x axis is really the
graph of y equals zero; it's that magic
point on the y axis called
zero and we want to get there. You might ask
why zero? Why not take y equals seven as
your goal? And we could do that too,
exactly the same proof would do that too.
I want to graph it, well unfortunately I
don't even know where to start. How can I
graph a function that I know so little
about? But I do know one thing: a positive
number to an odd power is positive, but a
negative number to an odd power is a
negative number. And if I make the
absolute value of the number really
really big then I get a really really
big number; and if the number is positive
it's a really really really positive
number, if it's negative it's a really
really negative. Let's say a and b and c
and all the constants of this are pretty small,
they're bounded by something anyway. And
I take x to be 100 times as big as
the biggest constant. Then x- and let's
say the exponent's 5 just for the sake of
argument - then x to the fifth power will
be 100 times larger than x to the
fourth power because I had- I multiplied
it an extra time by x. So this first term
will be enormously larger than anything
else. And in fact if I graph this
function you won't even see the
influence of the other- other terms on
the scale that I'm trying to draw here.
Basically this first term is going to
dominate, it's going to be the most
important term. And so I can at least
approximate what I'm doing,
especially when x is large, just by
thinking about the first term. Of course
when x is small all bets are off, I can't say a thing;
but when x is large I understand. So
let's take x large, let's take x
100 times larger than all the other
coefficients - I'll just pretend it's
100 for the sake of argument. - (So a big x.)
A big x, x big. And I'm going to say
absolute x big, and this is the way a
mathematician would write that, absolute
x much bigger than zero; so minus 100,
plus 100, those are good candidates. So if
x is big, x is 100, then x to this
positive power will be much bigger than
any of the other exponents and it'll be
about 100 to the 2n plus 1;
that's a big number.
I'm going to draw it up here and you
have to think that the y-axis is is a
really different scale than the x axis.
That point I can put in with some
confidence, I don't know exactly where it
is, pretty close. - (It's way out there in the-)
It's way out
there, right. Now if I take x to be minus
100 then in- for the same reason, now I
get enormous absolute value for x to the
fifth or x to the 2n plus 1; but 2n plus
1 is odd so I'll get an enormous
negative value. So that was the first
point of the proof was that I can- I can
say something about these way-out values.
The second point of the proof - and there
are only two - is that a graph of a
function like this, which is called a
continuous function, can be drawn without
lifting my pencil. So I'm going to start
drawing. I don't know where the graph
really goes but I'm going to draw as if
I did. It goes somehow. And when I get to
this point - oh my gosh - it has to cross
the x-axis to get up there. Well, all
right, here we go.
It just crossed, that's a root of the
equation; and it goes on somehow. I don't
have any idea really where the root is;
it might be on the left, it might be on
the right, but in order to get from here
to here without lifting your pencil you
simply have to draw a line that crosses.
(Like you can't walk from Mexico to Canada)
(without going through the US?) - Right, right,
of course you could swim around and you-
or you maybe who could dig a tunnel. And
so we might have a little worry - how do
we really know that there wasn't a
little tunnel just at that place? Maybe
the real numbers just opened up like the
Red Sea and let the red thing through?
Well, that's a really basic point about
the real numbers and it wasn't
completely understood until Dedekind I
think in the nineteenth century, but you
have to decide what you think the real
numbers are. If I took the rational
numbers- for instance, suppose I said I
don't know what the real numbers are but
let's do with the rational numbers instead;
this wouldn't have worked because
there- for example x cubed minus 2 has no real roots,
that's an odd degree polynomial, and
that's because this line would slip
through the rational numbers where
there's no rational number. And even
adding square roots and cube roots
wouldn't be enough, you'd have to add a
lot of roots to make
this work. So how do we know there isn't
a hole there? And this is one of the
great ideas in mathematics that's due to
Dedekind and it was a definition of
real numbers in terms of rational
numbers. And it's so nice that I have to
tell you about it even though you might
already be convinced that there's no
holes in the real numbers. So I'm going
to draw the rational numbers like this,
there are a lot of rational numbers but
they're also lots of holes as we've seen.
And Dedekind had the idea that what
a real number really is is a way of
dividing the rational numbers into two
parts. So here- let's say this is going to
be a real number,
a and Dedekind said that what a really
is is the set of rationals- rational
numbers less than a, together with the
set of rational numbers greater than a.
So these two sets tell where a is in the
real numbers. And the reason there are no
holes left is if there were a hole I
would just find- I would just define it
to be the number which is the two sets,
all the numbers the rational numbers
less than that number and all the
numbers greater than that number. So I've
defined holes out of existence this way.
(It's a really funny definition-) - It's a
great trick. - (That's like me saying the)
(definition of David is the left side of)
(his body plus the right side of his body.)
It's worse than that, it's saying it's
the space on the left of me together
with the space on the right of me; and it
tells you just as little about me as
this tells you about the real number
that's sitting there. But it's a- it's a
good existence kind of proof, or really
construction, this is the sort of thing
that people study in maybe the junior
year course in mathematics. - (Or philosophy by the sound of it.)
Or philosophy, yes. These are called
Dedekind cuts and it's a really important
idea in mathematics.
I want to graph this function, I want to
tell you what happens when I apply x and I
want to find a point here which - or at
least to convince you that there is a
point - a blue point which will go to zero
on the red side.